Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#564 ⋊ C2 |
| Sylow | C3 ⋊ #3553 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#564 | C2 |
| hypercenter | #564 | D3 |
| hypocenter | C3 | #3553 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #564 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2)(21,26)(25,28)(27,31)(34,35);(5,11)(9,18)(10,14)(17,19)(20,23,24,33)(21,27,26,31)(22,30,29,32)(25,34,28,35);(2,3)(4,7,8,16)(5,10,11,14)(6,12,13,15)(9,17,18,19)(21,26)(22,29)(27,31)(30,32);(2,3)(4,6,8,13)(5,9,11,18)(7,15,16,12)(10,19,14,17)(20,22,24,29)(21,25,26,28)(23,32,33,30)(27,35,31,34);(2,3)(4,5)(6,9)(7,14)(8,11)(10,16)(12,19)(13,18)(15,17)(20,21)(22,28)(23,31)(24,26)(25,29)(27,33)(30,34)(32,35)Polycyclic presentation:
2^7,3|b^2=g,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df,h^b=h^2,h^c=h^2,h^e=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10777055, #10777063, #10777094, … |
| maximal normal subgroup of | #10777055, #10777063, #10777094, … |
| maximal quotient of | #10958076, #10958310, #10958765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19719);